Problem 1. Circles in the Plane
… the Power of the World always works in circles, and everything tries to be round.
-- Black Elk in Black Elk Speaks (Neihardt)
In our study of circles we will need two criteria for similar triangles and a familiar theorem about isosceles triangles:
AAA criterion: If two triangles are similar (that is they have the same angles), then the corresponding sides of the triangles are in the same proportion to one another.
SAS criterion: If two triangles have an angle in common and if the corresponding sides of the angle are in the same proportion to each other, then the triangles have the same angles.
Isosceles Triangle Theorem: If a triangle has two sides equal in length then the two angles not shared by these two sides are equal.
Exterior Angle Theorem: The exterior angle of any triangle on the plane is equal to the sum of the other two interior angles.
Angles and Power Points of Plane Circles
a. If an arc of a circle subtends an angle 2α from the center of the circle, then the same arc subtends an angle α from any point on the circumference.
Use the figures. Draw a segment from the center of the circle to the point A and use the Isosceles Triangle Theorem and the Exterior Angle Theorem. Note the four different locations for A.
b. If two lines through a point P intersect a circle at points A, A′ (possibly coincident) and B, B′ (possibly coincident), then
|PA| × |PA′| = |PB| × |PB′|
This product is called the power of the point P with respect to the circle.
Use the figures below and draw the segment joining A to B′ and the segment joining A′ to B. Then apply Part a, and look for similar triangles.
Problem 1. Circles in the Plane, which explores some geometry of circles.
Problem 2. Inversions in Circles, which explores properties of inversion in circles.
Problem 3. Applications of Inversions to the Peaucellier-Lipkin Linkage, which explores the applications of Problems 1 and 2 to an understanding of the linkage.